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Chapter 3: Q7P (page 159)

Generalize Problem 6 to three dimensions; to n dimensions.

Short Answer

Expert verified

$C$ is an orthogonal matrix.

Step by step solution

01

Given information

C is a matrix whose columns are the components $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ of two perpendicular vectors, each of unit length.

02

Orthogonal matrix

An orthogonal matrix, also known as an orthonormal matrix, is a real square matrix with orthonormal vectors in the columns and rows.

03

Calculate matrix C

To generalize matrix $C$ , we need to represent n-dimensional matrix. Therefore, we will need $n$ orthogonal vectors $r_{n}$ each with $n$ components.

A n-dimensional matrix can be mathematically presented as

$C = (r_{1} r_{2} r_{2} . . . . . . . r_{n}), w h e r e r_{i}^{T} r_{j} = δ_{i j}, a n d δ_{i j} = \{\begin{matrix} 1 \\ 0 \end{matrix} \begin{matrix} i = j \\ i \neq j \end{matrix}$

Now, calculate $C^{T} C$ .

$C^{T} C = (\begin{matrix} r_{1}^{T} \\ r_{2}^{T} \\ ⋰ \\ . . \\ r_{n}^{T} \end{matrix}) (r_{1} r_{2} r_{2} . . . . . . . . . r_{n})$

Therefore, ${(C^{T} C)}_{i j} = r_{i}^{T} r_{j} = δ_{i j}$ , which shows that $C$ is an orthogonal matrix.

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Most popular questions from this chapter

Verify formula (6.13). Hint: Consider the product of the matrices ${MC}^{T}$ . Use Problem 3.8.

Let each of the following matrices Mdescribe a deformation of the $(x, y)$ plane For each given M find: the Eigen values and eigenvectors of the transformation, the matrix Cwhich DiagonalizesM and specifies the rotation to new axes $(x', y')$ along the eigenvectors, and the matrix D which gives the deformation relative to the new axes. Describe the deformation relative to the new axes.

$(\begin{matrix} 3 & 4 \\ 4 & 9 \end{matrix})$

Verify that each of the following matrices is Hermitian. Find its eigenvalues and eigenvectors, write a unitary matrix U which diagonalizes H by a similarity transformation, and show that $U^{- 1} H U$ is the diagonal matrix of eigenvalues.

$(\begin{matrix} 2 & i \\ - i & 2 \end{matrix})$

Find the Eigen values and eigenvectors of the following matrices. Do some problems by hand to be sure you understand what the process means. Then check your results by computer

$(\begin{matrix} 1 & 2 & 2 \\ 2 & 3 & 0 \\ 2 & 0 & 3 \end{matrix})$

Show that the product $A A^{Τ}$ is a symmetric matrix.

See all solutions

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